文摘
It was shown recently by Conti, Rørdam and Szymański that there exist endomorphisms class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301926&_mathId=si1.gif&_user=111111111&_pii=S0022123616301926&_rdoc=1&_issn=00221236&md5=5a9d11d6d965b406a5956f4ad053d26f" title="Click to view the MathML source">λuclass="mathContainer hidden">class="mathCode"> of the Cuntz algebra class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301926&_mathId=si2.gif&_user=111111111&_pii=S0022123616301926&_rdoc=1&_issn=00221236&md5=aca49272aa3088c7f945ffcb120c431f" title="Click to view the MathML source">Onclass="mathContainer hidden">class="mathCode"> such that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301926&_mathId=si3.gif&_user=111111111&_pii=S0022123616301926&_rdoc=1&_issn=00221236&md5=e9354c2a92d55bce9025ad0f7e3258be" title="Click to view the MathML source">λu(Fn)⊆Fnclass="mathContainer hidden">class="mathCode"> but class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301926&_mathId=si4.gif&_user=111111111&_pii=S0022123616301926&_rdoc=1&_issn=00221236&md5=7a2e9c6326526cd2d4c8e81e39d07958" title="Click to view the MathML source">u∉Fnclass="mathContainer hidden">class="mathCode">, and a question was raised if for such a u there must always exist a unitary class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301926&_mathId=si202.gif&_user=111111111&_pii=S0022123616301926&_rdoc=1&_issn=00221236&md5=adeb6c6193360672f98ec12c80b8ec0f" title="Click to view the MathML source">v∈Fnclass="mathContainer hidden">class="mathCode"> with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301926&_mathId=si203.gif&_user=111111111&_pii=S0022123616301926&_rdoc=1&_issn=00221236&md5=4c7bfa33c4962aba98ffbadf6381fa6d" title="Click to view the MathML source">λu|Fn=λv|Fnclass="mathContainer hidden">class="mathCode">. In the present paper, we answer this question to the negative. To this end, we analyze the structure of such endomorphisms class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301926&_mathId=si1.gif&_user=111111111&_pii=S0022123616301926&_rdoc=1&_issn=00221236&md5=5a9d11d6d965b406a5956f4ad053d26f" title="Click to view the MathML source">λuclass="mathContainer hidden">class="mathCode"> for which the relative commutant class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301926&_mathId=si108.gif&_user=111111111&_pii=S0022123616301926&_rdoc=1&_issn=00221236&md5=39f15958d000c9489b14fb1d4b744155" title="Click to view the MathML source">λu(Fn)′∩Fnclass="mathContainer hidden">class="mathCode"> is finite dimensional.